Rasterization. MIT EECS Frédo Durand and Barb Cutler. MIT EECS 6.837, Cutler and Durand 1

Transcription

1 Rasterization MIT EECS Frédo Durand and Barb Cutler MIT EECS 6.837, Cutler and Durand 1

2 Final projects Rest of semester Weekly meetings with TAs Office hours on appointment This week, with TAs Refine timeline Define high-level architecture Project should be a whole, but subparts should be identified with regular merging of code MIT EECS 6.837, Cutler and Durand 2

3 The Graphics Pipeline Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display MIT EECS 6.837, Cutler and Durand 3

4 Modeling Transformations Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping 3D models defined in their own coordinate system (object space) Modeling transforms orient the models within a common coordinate frame (world space) Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display Object space World space MIT EECS 6.837, Cutler and Durand 4

5 Illumination (Shading) (Lighting) Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Vertices lit (shaded) according to material properties, surface properties (normal) and light sources Local lighting model (Diffuse, Ambient, Phong, etc.) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display MIT EECS 6.837, Cutler and Durand 5

6 Viewing Transformation Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Maps world space to eye space Viewing position is transformed to origin & direction is oriented along some axis (usually z) z o y x WORLD SPACE FAR -n NEAR o P Image adapted from: Seth Teller u v EYE EYE SPACE Scan Conversion (Rasterization) Visibility / Display MIT EECS 6.837, Cutler and Durand 6

7 Clipping Modeling Transformations Illumination (Shading) Transform to Normalized Device Coordinates (NDC) Portions of the object outside the view volume (view frustum) are removed Viewing Transformation (Perspective / Orthographic) Clipping z FAR o NEAR x y EYE z x y Projection (to Screen Space) EYE SPACE Image adapted from: Seth Teller NDC Scan Conversion (Rasterization) Visibility / Display MIT EECS 6.837, Cutler and Durand 7

8 Projection Modeling Transformations The objects are projected to the 2D image place (screen space) Illumination (Shading) x y x y Viewing Transformation (Perspective / Orthographic) z z o Clipping Projection (to Screen Space) Scan Conversion (Rasterization) NDC Image adapted from: Seth Teller SCREEN SPACE Visibility / Display MIT EECS 6.837, Cutler and Durand 8

9 Scan Conversion (Rasterization) Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Rasterizes objects into pixels Interpolate values as we go (color, depth, etc.) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display MIT EECS 6.837, Cutler and Durand 9

10 Visibility / Display Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display Each pixel remembers the closest object (depth buffer) Almost every step in the graphics pipeline involves a change of coordinate system. Transformations are central to understanding 3D computer graphics. MIT EECS 6.837, Cutler and Durand 10

11 Today Polygon scan conversion smart back to brute force Visibility MIT EECS 6.837, Cutler and Durand 11

12 2D Scan Conversion Geometric primitive 2D: point, line, polygon, circle... 3D: point, line, polyhedron, sphere... Primitives are continuous; screen is discrete MIT EECS 6.837, Cutler and Durand 12

13 2D Scan Conversion Solution: compute discrete approximation Scan Conversion: algorithms for efficient generation of the samples comprising this approximation MIT EECS 6.837, Cutler and Durand 13

14 Brute force solution for triangles? MIT EECS 6.837, Cutler and Durand 14

15 Brute force solution for triangles For each pixel Compute line equations at pixel center clip against the triangle Problem? MIT EECS 6.837, Cutler and Durand 15

16 Brute force solution for triangles For each pixel Compute line equations at pixel center clip against the triangle Problem? If the triangle is small, a lot of useless computation MIT EECS 6.837, Cutler and Durand 16

17 Brute force solution for triangles Improvement: Compute only for the screen bounding box of the triangle How do we get such a bounding box? MIT EECS 6.837, Cutler and Durand 17

18 Brute force solution for triangles Improvement: Compute only for the screen bounding box of the triangle Xmin, Xmax, Ymin, Ymax of the triangle vertices MIT EECS 6.837, Cutler and Durand 18

19 Can we do better? Kind of! We compute the line equation for many useless pixels What could we do? MIT EECS 6.837, Cutler and Durand 19

20 Use line rasterization Compute the boundary pixels Shirley page 55 MIT EECS 6.837, Cutler and Durand 20

21 Scan-line Rasterization Compute the boundary pixels Fill the spans Shirley page 55 MIT EECS 6.837, Cutler and Durand 21

22 Scan-line Rasterization Requires some initial setup to prepare Shirley page 55 MIT EECS 6.837, Cutler and Durand 22

23 Clipping problem How do we clip parts outside window? MIT EECS 6.837, Cutler and Durand 23

24 Clipping problem How do we clip parts outside window? Create two triangles or more. Quite annoying. MIT EECS 6.837, Cutler and Durand 24

25 Old style graphics hardware Triangles were big Bresenham+interpolation is worth it Annoying clipping step + a couple of other things have changed Not that good to parallelize beyond triangle granularity MIT EECS 6.837, Cutler and Durand 25

26 Questions? MIT EECS 6.837, Cutler and Durand 26

27 Today Polygon scan conversion smart back to brute force Visibility MIT EECS 6.837, Cutler and Durand 27

28 For modern graphics cards Triangles are usually very small Setup cost are becoming more troublesome Clipping is annoying Brute force is tractable MIT EECS 6.837, Cutler and Durand 28

29 Modern rasterization For every triangle ComputeProjection Compute bbox, clip bbox to screen limits For all pixels in bbox Compute line equations If all line equations>0 //pixel [x,y] in triangle Framebuffer[x,y]=triangleColor MIT EECS 6.837, Cutler and Durand 29

30 Modern rasterization For every triangle ComputeProjection Compute bbox, clip bbox to screen limits For all pixels in bbox Compute line equations If all line equations>0 //pixel [x,y] in triangle Framebuffer[x,y]=triangleColor Note that Bbox clipping is trivial MIT EECS 6.837, Cutler and Durand 30

31 Can we do better? For every triangle ComputeProjection Compute bbox, clip bbox to screen limits For all pixels in bbox Compute line equations If all line equations>0 //pixel [x,y] in triangle Framebuffer[x,y]=triangleColor MIT EECS 6.837, Cutler and Durand 31

32 Can we do better? For every triangle ComputeProjection Compute bbox, clip bbox to screen limits For all pixels in bbox Compute line equations ax+by+c If all line equations>0 //pixel [x,y] in triangle Framebuffer[x,y]=triangleColor We don t need to recompute line equation from scratch MIT EECS 6.837, Cutler and Durand 32

33 Can we do better? For every triangle ComputeProjection Compute bbox, clip bbox to screen limits Setup line eq compute a i dx, b i dy for the 3 lines Initialize line eq, values for bbox corner L i =a i x0+b i y+c i For all scanline y in bbox For 3 lines, update Li For all x in bbox Increment line equations: Li+=adx If all Li>0 //pixel [x,y] in triangle Framebuffer[x,y]=triangleColor We save one multiplication per pixel MIT EECS 6.837, Cutler and Durand 33

34 The Graphics Pipeline Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Modern hardware mostly avoids clipping Only with respect to plane z=0 Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display MIT EECS 6.837, Cutler and Durand 34

35 Full Clipping "clip" geometry to view frustum (eye x, eye y, eye z ) z axis image plane MIT EECS 6.837, Cutler and Durand 35

36 One-plane clipping "clip" geometry to near plane (eye x, eye y, eye z ) z axis image plane MIT EECS 6.837, Cutler and Durand 36

37 Adding Gouraud shading Interpolate colors of the 3 vertices Linear interpolation MIT EECS 6.837, Cutler and Durand 37

38 Adding Gouraud shading Interpolate colors of the 3 vertices Linear interpolation, e.g. for R channel: R=a R x+b R y+c R Such that R[x0,y0]=R0; R[x1, y1]=r1; R[x2,y2]=R2 Same as a plane equation in (x,y,r) MIT EECS 6.837, Cutler and Durand 38

39 Adding Gouraud shading Interpolate colors For every triangle ComputeProjection Compute bbox, clip bbox to screen limits Setup line eq Setup color equation For all pixels in bbox Increment line equations Increment color equation If all Li>0 //pixel [x,y] in triangle Framebuffer[x,y]=interpolatedColor MIT EECS 6.837, Cutler and Durand 39

40 In the modern hardware Edge eq. in homogeneous coordinates [x, y, w] Tiles to add a mid-level granularity early rejection of tiles Memory access coherence MIT EECS 6.837, Cutler and Durand 40

41 Ref Henry Fuchs, Jack Goldfeather, Jeff Hultquist, Susan Spach, John Austin, Frederick Brooks, Jr., John Eyles and John Poulton, Fast Spheres, Shadows, Textures, Transparencies, and Image Enhancements in Pixel-Planes, Proceedings of SIGGRAPH 85 (San Francisco, CA, July 22 26, 1985). In Computer Graphics, v19n3 (July 1985), ACM SIGGRAPH, New York, NY, Juan Pineda, A Parallel Algorithm for Polygon Rasterization, Proceedings of SIGGRAPH 88 (Atlanta, GA, August 1 5, 1988). In Computer Graphics, v22n4 (August 1988), ACM SIGGRAPH, New York, NY, Figure 7: Image from the spinning teapot performance test. Triangle Scan Conversion using 2D Homogeneous Coordinates, Marc Olano Trey Greer MIT EECS 6.837, Cutler and Durand 41

42 Take-home message The appropriate algorithm depends on Balance between various resources (CPU, memory, bandwidth) The input (size of triangles, etc.) Smart algorithms often have initial preprocess Assess whether it is worth it To save time, identify redundant computation Put outside the loop and interpolate if needed MIT EECS 6.837, Cutler and Durand 42

43 Today Polygon scan conversion smart back to brute force Visibility MIT EECS 6.837, Cutler and Durand 43

44 Visibility How do we know which parts are visible/in front? MIT EECS 6.837, Cutler and Durand 44

45 Ray Casting Maintain intersection with closest object MIT EECS 6.837, Cutler and Durand 45

46 Painter s algorithm Draw back-to-front How do we sort objects? Can we always sort objects? MIT EECS 6.837, Cutler and Durand 46

47 Painter s algorithm Draw back-to-front How do we sort objects? Can we always sort objects? No, there can be cycles Requires to split polygons A B A C B MIT EECS 6.837, Cutler and Durand 47

48 Painter s algorithm Old solution for hidden-surface removal But Good because ordering is useful for other operations (transparency, antialiasing) Ordering is tough Cycles Must be done by CPU Hardly used now But some sort of partial ordering is sometimes useful Usuall front-to-back To make sure foreground is rendered first MIT EECS 6.837, Cutler and Durand 48

49 visibility In ray casting, use intersection with closest t Now we have swapped the loops (pixel, object) How do we do? MIT EECS 6.837, Cutler and Durand 49

50 Z buffer In addition to frame buffer (R, G, B) Store distance to camera (z-buffer) Pixel is updated only if new z is closer than z-buffer value MIT EECS 6.837, Cutler and Durand 50

51 Z-buffer pseudo code For every triangle Compute Projection, color at vertices Setup line equations Compute bbox, clip bbox to screen limits For all pixels in bbox Increment line equations Compute curentz Increment currentcolor If all line equations>0 //pixel [x,y] in triangle If currentz<zbuffer[x,y] //pixel is visible Framebuffer[x,y]=currentColor zbuffer[x,y]=currentz MIT EECS 6.837, Cutler and Durand 51

52 Works for hard cases! A C B MIT EECS 6.837, Cutler and Durand 52

53 What exactly do we store Floating point distance Can we interpolate z in screen space? i.e. does z vary linearly in screen space? z0 image x x z1 MIT EECS 6.837, Cutler and Durand 53

54 Z interpolation X =x/z Hyperbolic variation Z cannot be linearly interpolated z0 image x x z1 MIT EECS 6.837, Cutler and Durand 54

55 Simple Perspective Projection Project all points along the z axis to the z = d plane, eyepoint at the origin homogenize x * d / z y * d / z d 1 = x x y y = z z z / d 0 0 1/d MIT EECS 6.837, Cutler and Durand

56 Yet another Perspective Projection Change the z component Compute d/z Can be linearly interpolated homogenize x * d / z y * d / z d/z 1 = x x y y = z z / d 0 0 1/d MIT EECS 6.837, Cutler and Durand

57 Advantages of 1/z Can be interpolated linearly in screen space Puts more precision for close objects Useful when using integers more precision where perceptible MIT EECS 6.837, Cutler and Durand 57

58 Integer z-buffer Use 1/z to have more precision in the foreground Set a near and far plane 1/z values linearly encoded between 1/near and 1/far Careful, test direction is reversed far near x MIT EECS 6.837, Cutler and Durand 58

59 Integer Z-buffer pseudo code For every triangle Compute Projection, color at vertices Setup line equations, depth equation Compute bbox, clip bbox to screen limits For all pixels in bbox Increment line equations Increment curent_1ovz Increment currentcolor If all line equations>0 //pixel [x,y] in triangle If current_1ovz>1ovzbuffer[x,y]//pixel is visible Framebuffer[x,y]=currentColor 1ovzBuffer[x,y]=current1ovZ MIT EECS 6.837, Cutler and Durand 59

60 Gouraud interpolation Gouraud: interpolate color linearly in screen space Is it correct? [R0, z0 G0, B0] image x x [R1, z1 G1, B1] MIT EECS 6.837, Cutler and Durand 60

61 Gouraud interpolation Gouraud: interpolate color linearly in screen space Not correct. We should use hyperbolic interpolation But quite costly (division) [R0, z0 G0, B0] image x x [R1, z1 G1, B1] MIT EECS 6.837, Cutler and Durand 61

62 Next time Clipping Segment intersection & scanline algorithms MIT EECS 6.837, Cutler and Durand 62